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Nonconvex Optimization and Its Applications

A new decimation scheme for the symbolic dynamics of the Feigenbaum attractors of unimodal maps is established. The coarse-grained statistical properties of the attractors are subsequently derived, with emphasis on the behavior of the block entropies.

Symbolic Dynamics and Entropy Analysis of Feigenbaum Limit Sets

A coupled map lattice is introduced that simulates the time evolution of velocity differences in fully developed turbulent flows. The model considered is an extension of the Langevin theory to chaotic driving forces acting on a self-similar cascade of spatial levels. Compared to full simulations of the Navier-Stokes equation, the amount of necessary computing time is negligible. Despite its simplicity, the model is in perfect agreement with experimentally observed results, provided the chaotic driving force is generated by the fully developed logistic map with parameter value \ensuremath{\mu}=2. The shape of the velocity distributions, the slight asymmetry, the stretched exponential tails, as well as the moment scaling exponents ${\mathrm{\ensuremath{\zeta}}}_{\mathit{m}}$, come out in precisely the same way as in experimental measurements of high Reynolds number flows.

Chaotic cascade model for turbulent velocity distributions

We review recent progress on Feigenbaum attractors and their interconnection with Number Theory. We further enlight the relation between Chaos and Transcendence.

From Symbolic Dynamics to a Digital Approach: Chaos and Transcendence

We study resonances of nonlinear systems of differential equations, including but not limited to the equations of motion of a particle moving in a potential. We use the calculus of variations to determine the minimal additive forcing function that induces a desired terminal response, such as an energy in the case of a physical system. We include the additional constraint that only select degrees of freedom be forced, corresponding to a very general class of problems in which not all of the degrees of freedom in an experimental system are accessible to forcing. We find that certain Lagrange multipliers take on a fundamental physical role as the effective forcing experienced by the degrees of freedom which are not forced directly. Furthermore, we find that the product of the displacement of nearby trajectories and the effective total forcing function is a conserved quantity. We demonstrate the efficacy of this methodology with several examples.

Resonant forcing of nonlinear systems of differential equations.

The dynamics of bistable oscillators driven by periodic dichotomous noise is described The stochastic differential equation governing the flow implies smooth trajectories between noise switching events The dynamics of the two-branched map induced by this flow is a Markov process Harmonic and quartic models of the bistable potential are studied in the overdamped limit In the linear (harmonic) case the dynamics can be reduced to a stochastic one-dimensional map with two branches The moments decay exponentially in this case, although the invariant measure may be multifractal For strong damping, relaxation induces a cascade leading to a Cantor set and anomalous decay of the density in this case is modeled by a Markov chain For the physically more realistic case of a quartic potential many additional features arise since the contraction factor is distance dependent By tuning the barrier-height parameter in the quartic potential, noise-induced transition rates with the characteristics of intermittency are found

Dynamics of oscillators with periodic dichotomous noise

Attractor structure on the real line is represented as a mass function M(x) defined as the fraction of attractor points to the left of x. For Feigenbaum attractors, M is a devil's staircase which possesses both orbit and mass scaling features. The attractors ${\mathrm{scrA}}_{p}$ for maps with a pth-order extremum are discussed in terms of this scaling, which leads to a hierarchy of bounds for the corresponding mass ${M}_{p}$. Both ${\mathrm{scrA}}_{p}$ and ${M}_{p}$ transform in a characteristic way under power-law conjugacy; this gives rise to two kinds of global scaling. A further intermediate scaling for the quartic attractor which underlies vector scaling in two-extremum maps is discussed in this context. As p tends to infinity, the capacity dimension D tends to 1 and the information dimension \ensuremath{\sigma} to 0. The effects of this trend on the appearance and observability of attractors are discussed.

Mass and dimension of Feigenbaum attractors.

Space-time lattice (cellular automaton) models of pattern formation and growth are described. Suitable local rules for automation evolution represent the spreading of wave fronts of activity in an excitable medium. A random distribution of seeds produces expanding rings that fuse and are annihilated. The seeding density, pA, is used as a scaling parameter to give unique, reduced dynamics in an arbitrary dimension d. For d = 2, in this (continuum) picture, the rings fuse globally (percolate) at a critical instant, tc = 0.45. For the unscaled time evolution, dynamical percolation is examined in the pA × t plane. A swath of these percolating states is found. On the “explosion” boundary of this swath the percolation cluster just forms; on the “implosion” boundary it breaks up. Using a small-sample method the fractal dimension of the critical (boundary) cluster is estimated to be 1.9 (±0.01). Also percolation for continuously emitting seeds, which produce “discs” of activity, is related to ring evolution.

Discrete models of growth and dynamical percolation in chemistry

The general formalism for periodic dichotomous noise on nonpotential flows is considered. This uncorrelated noise process switches suddenly at integer values of period $\ensuremath{\tau}$. The effect of additive noise of this kind on the planar FitzHugh-Nagumo ordinary differential equations [R. FitzHugh, Biophys. J. 1, 445 (1961); J. Nagumo, S. Arimoto, and Y. Yoshikawa, Proc. IRE 50, 2061 (1962)] is examined. For large $\ensuremath{\tau}$, quasifractal attractors are observed, whereas for the white-noise limit, where $\ensuremath{\tau}$ is small, a Fokker-Planck equation describes the evolution. The magnitude of $\ensuremath{\tau}$ determines the smoothness of the transient evolution and equilibrium density of the system. Typically the stochastic equations give rise to two regions of high density near the stable fixed points of the underlying autonomous system. The stiffness parameter $\ensuremath{\varepsilon}$ in the differential equations determines the fast variable, its associated nullcline, and the resulting flow structure. For small $\ensuremath{\varepsilon}$ the cubic nullcline controls the motion and transitions between the high-density peaks occur along segments of a noisy limit cycle. For large $\ensuremath{\varepsilon}$ the linear nullcline governs the transitions and the peaks are joined by a single band. The statistical behavior of the oscillatory and direct transitions is examined.

Dichotomously switched phase flows

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Papers

Dynamics of oscillators with periodic dichotomous noise

Mass and dimension of Feigenbaum attractors.

Discrete models of growth and dynamical percolation in chemistry

Dichotomously switched phase flows